An adaptive finite difference method using far-field boundary conditions for the Black--Scholes equation
Bull. Korean Math. Soc. 2014 Vol. 51, No. 4, 1087-1100
Printed July 31, 2014
Darae Jeong, Taeyoung Ha, Myoungnyoun Kim, Jaemin Shin, In-Han Yoon, and Junseok Kim
Korea University, National Institute for Mathematical Sciences, National Institute for Mathematical Sciences, Ewha Womans University, Korea University, Korea University
Abstract : We present an accurate and efficient numerical method for solving the Black--Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.
Keywords : Black--Scholes equation, finite difference method, far-field boundary conditions, adaptive grid, Peclet condition
MSC numbers : Primary 65M06, 65M50
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